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Parikh vectors
3AOB_1 1ZWG_1 5TAM_1 Letter Amino acid
47 1 5 D Aspartic acid
48 2 5 Q Glutamine
9 1 0 W Tryptophan
27 0 2 Y Tyrosine
105 1 6 A Alanine
50 3 7 E Glutamic acid
89 1 13 G Glycine
42 2 3 M Methionine
45 0 7 P Proline
67 0 8 T Threonine
97 3 7 V Valine
2 0 2 C Cysteine
15 3 2 H Histidine
72 1 6 I Isoleucine
38 2 5 R Arginine
39 3 4 N Asparagine
102 6 7 L Leucine
43 3 9 K Lycine
54 1 6 F Phenylalanine
62 1 4 S Serine 

3AOB_1|Chains A, B, C|Acriflavine resistance protein B|Escherichia coli (83333)
>1ZWG_1|Chain A|PARATHYROID HORMONE|Homo sapiens (9606)
>5TAM_1|Chains A, B[auth F], C[auth H], D[auth J]|Peptidyl-prolyl cis-trans isomerase FKBP1B|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
3AOB , Knot 376 1053 0.82 40 301 937 
MPNFFIDRPIFAWVIAIIIMLAGGLAILKLPVAQYPTIAPPAVTISASYPGADAKTVQDTVTQVIEQNMNGIDNLMYMSSNSDSTGTVQITLTFESGTDADIAQVQVQNKLQLAMPLLPQEVQQQGVSVEKSSSSFLMVVGVINTDGTMTQEDISDYVAANMKDAISRTSGVGDVQLFGSQYAMRIWMNPNELNKFQLTPVDVITAIKAQNAQVAAGQLGGTPPVKGQQLNASIIAQTRLTSTEEFGKILLKVNQDGSRVLLRDVAKIELGGENYDIIAEFNGQPASGLGIKLATGANALDTAAAIRAELAKMEPFFPSGLKIVYPYDTTPFVKISIHEVVKTLVEAIILVFLVMYLFLQNFRATLIPTIAVPVVLLGTFAVLAAFGFSINTLTMFGMVLAIGLLVDDAIVVVENVERVMAEEGLPPKEATRKSMGQIQGALVGIAMVLSAVFVPMAFFGGSTGAIYRQFSITIVSAMALSVLVALILTPALCATMLKPIAKGDHGEGKKGFFGWFNRMFEKSTHHYTDSVGGILRSTGRYLVLYLIIVVGMAYLFVRLPSSFLPDEDQGVFMTMVQLPAGATQERTQKVLNEVTHYYLTKEKNNVESVFAVNGFGFAGRGQNTGIAFVSLKDWADRPGEENKVEAITMRATRAFSQIKDAMVFAFNLPAIVELGTATGFDFELIDQAGLGHEKLTQARNQLLAEAAKHPDMLTSVRPNGLEDTPQFKIDIDQEKAQALGVSINDINTTLGAAWGGSYVNDFIDRGRVKKVYVMSEAKYRMLPDDIGDWYVRAADGQMVPFSAFSSSRWEYGSPRLERYNGLPSMEILGQAAPGKSTGEAMELMEQLASKLPTGVGYDWTGMSYQERLSGNQAPSLYAISLIVVFLCLAALYESWSIPFSVMLVVPLGVIGALLAATFRGLTNDVYFQVGLLTTIGLSAKNAILIVEFAKDLMDKEGKGLIEATLDAVRMRLRPILMTSLAFILGVMPLVISTGAGSGAQNAVGTGVMGGMVTATVLAIFFVPVFFVVVRRRFSRKNEDIEHSHTVDHHHHHH
1ZWG , Knot 24 35 0.81 34 33 33 
XEIQLMHNLGKHLNSMERVEWLRKKLQDVHNFVAL
5TAM , Knot 57 108 0.82 38 92 106 
MGVEIETISPGDGRTFPKKGQTCVVHYTGMLQNGKKFDSSRDRNKPFKFRIGKQEVIKGFEEGAAQMSLGQRAKLTCTPDVAYGATGHPGVIPPNATLIFDVELLNLE

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(3AOB_1)}(2) \setminus P_{f(1ZWG_1)}(2)|=275\), \(|P_{f(1ZWG_1)}(2) \setminus P_{f(3AOB_1)}(2)|=7\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:110111001111111111111111111101111001011111101010011101001000100110001011001101000000010101010100100101101010001011111110010001101000000111111110001010000100011101001100001110101110001101110100100101011011011010010111101110111010010101110001000001101110100010011100110101110000111010101101111011011011001111010110101111011011010000111010100110011011111111101110010101110111111111011111111101001011111111111100111110010011100111100100001101011111111110111111111110011100010101101111011111110111010110111010010100111111001100000000001111100010011101111111101110110011100001111011011111000000011001000010000001001111011111101000111110100110011000010110101001100100111111011111011010110101100111100010010001110110010110010101100010101010000101111010010001111111001001100101001011001000111001101010110101111011000010010101000011101011101111000101101100110011011100101100000101001101011011111101111000101110111111111111111101011000101011110011101001111101100110001011101010110101011110011111111111100111011001110111111101011111111111111000100000010000010000000
Pair \(Z_2\) Length of longest common subsequence
3AOB_1,1ZWG_1 282 3
3AOB_1,5TAM_1 223 4
1ZWG_1,5TAM_1 101 2

Newick tree

 
[
	3AOB_1:14.84,
	[
		5TAM_1:50.5,1ZWG_1:50.5
	]:93.34
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{1088 }{\log_{20} 1088}-\frac{35}{\log_{20}35})=296.\)
Status Protein1 Protein2 d d1/2
Query variables 3AOB_1 1ZWG_1 366 189
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]